In this paper, the novel design of a side-slip angle observer and an observer-based stability controller for a vehicle with a non-linear tyre model and a varying forward speed is presented. The Takagi-Sugeno fuzzy modelling technique is first applied to represent the vehicle's lateral dynamics with the non-linear Dugoff tyre model and varying speed. The observer and the observer-based controller are then constructed using the measured yaw rate and the estimated premise and state variables and are designed to be robust against the model and parameter uncertainties. The conditions for designing such an observer and an observer-based controller are derived in terms of linear matrix inequalities. Numerical simulations on a non-linear eight-degree-of-freedom vehicle dynamics model and experimental data are conducted to validate the effectiveness of the proposed approach. The comparison of results with those from other existing approaches shows that the designed observer can accurately estimate the vehicle's side-slip angle and the controller can effectively control this side-slip angle regardless of the variation in the vehicle's longitudinal velocity.